You were right on the money with your original result, but you were had by the "pounds mass" foolishness! There is no such thing as a pound mass. (That will bring them all down on my head, I'm sure, but it is true, unless you are willing to measure force in poundals.)

As long as force is in pounds, then pounds is only force. When force is in pounds, then there are two choices for mass, depending on the choice for length.

If lenght is in feet, then mass is in slugs = lb-s^2/ft.

If length is in inches, then mass is in lb-s^2/in, a unit that does not have a name (although the name "snail" was once suggested), but it is a perfectly good mass unit.

With either of these choices, Newton's law works just fine: F = m a, with no need for g sub c, or any other adjustments. It is bad engineering practice to make the mistake of using "pounds mass." A pound is a force!